Principal Investigator: Gloria Mari-Beffa
In this project the PI proposes to research the existence of geometric realizations for nonlinear biHamiltonian PDEs and the implications of this existence for the biHamiltonian system and for the realizing manifold alike. In particular she intends to describe possible realizing manifolds and to link their geometry to the type of system they realize. In the past the PI has studied how biHamiltonian structures are generated by the geometry of curves in realizing manifolds. She also studied geometric realizations of equations of KdV type in Hermitian Symmetric manifolds, linking the projective character of differential invariants to equations of KdV-type. In this project she proposes to deepen this relation and to expand it to parabolic manifolds. In joint work with M. Eastwood she will be using tools in classical differential geometry to find projective structures on flows. Resolving this differential geometry problem will very likely create geometric realizations of KdV-type in parabolic manifolds. She will also investigate the possibility of a similar connection in other geometries, for example that of Schrödinger, mKdV and sine-Gordon flows to Riemannian geometry. Finally, in joint work with Calini and Ivey, the PI will study geometric and topological properties of solutions to realizations of soliton equations, in particular those corresponding to finite-gap solutions (periodic case). The study will try to link properties of these solutions to their spectral parameter.
Setting an appropriate geometric background is often a fundamental step in the resolution of a problem. A choice of geometry establishes the properties and laws we wish to keep unchanged: Relativity (with Lorentzian geometry setting the interaction between space and time) and computer imaging (using projective geometry when 3D perspective needs to be preserved) are some of the best-known examples. But nowadays many engineers and physicists, including some groups working on data collection, consider a basic knowledge of differential geometry to be fundamental. Geometric thought is commonplace, as often finding the right choice of geometry for a problem is an initial step in its resolution - it all depends on how (or with which geometric eyes) you look at it -. BiHamiltonian nonlinear equations are very rich in structure and they are often used to model different types of phenomena. Their rich structure allows us to find a great deal of information about the system they model and to predict behavior. The best-known completely integrable systems are bi-Hamiltonian, and their solutions predict the behavior of fluids, from water waves in shallow water to the trailing vortices behind the wing tips of an airplane. These phenomena do not, in principle, exist within any given geometric background. When we find geometric realizations for a completely integrable system we gain information in two different ways: 1) We learn that the behavior of these phenomena can be visualized within a certain geometry, and we learn how to visualize it. In particular, the same phenomena can be described in more than one geometry; 2) We learn that some geometries are hosts to phenomena that were not known to exist in that context before. For example, by linking projective and centro-equi-affine geometries, we can find evolutions of star-shaped curves that behave like solitary waves. The better we understand the relation between these two apparently unrelated subjects, the more we can transfer our extensive geometric knowledge, their connections and properties, to the understanding of completely integrable systems. And vice-versa.
Intellectual merits: During the last four years (three years plus a small extension) I have worked on the interrelation of the moduli space of curves (curves described by their curvatures) and soliton equations (equations whose solutions are extremely stable, like smoke rings). In particular, I aimed to identify evolutions of curves in many different geometric backgrounds, each one with a different practical relevance, that will be soliton equations when written in terms of the curvatures. I also tied projective-type of curvatures to KDV-type of equations modelling waves. During the last two years the project moved into discretizations of these systems, a move that made it much more applicable to real life since everything in life - and certainly anything one enters in a computer - is discrete, and hence not really a curve but a polygon. Thus, we have aimed to define discrete moving frames in general geometries - frames that would allow us to study the properties and curvatures of polygons - not curves, and to apply them to the study of completely integrable discrete systems. The change was motivated by the study of the pentagram map, a map that associates to a polygon the polygon obtained by joining every other vertez of the polygon with a segment. Nevertheless because of its relevance when dealing with symmetries, we expect it to be important to some applications, like image analysis. Broader impact: the PI has organized a mentorship program for female high school students during 2004-2012. It served 7-15 students from local high schools each year, some of them minorities, working in math at three levels. The students work with graduates and undergraduates in math and have regular contact with female faculty. The PI tries to familiarize students with the world of research in mathematics. Simpler problems within this project were appropriate for undergraduate research and as Honors adviser in Madison the PI interacted regularly with undergraduates. During 2010-11 she supervised the undergraduate Honors thesis of Tess Anderson, and they co-authored a paper that was published in the Journal of Physics A, and included in the IOP "Select collection" ("Articles from the last 12 months that have been chosen by our editors for their novelty, significance and potential impact on future research"). Ms. Anderson is currently a graduate student at Brown. The PI also worked with Laura Balzano, a graduate student in Electrical and Computer Engineering who studied the application of Grassmannian techniques to the study of data. The PI helped her understand some relevant literature and how to do analysis on Grassmannians. Dr. Balzano has now a tenure-track position in Ann Arbor. At the other end of the spectrum, the PI is Chair of the committee remodeling pre-Calculus and Calculus classes in Madison, and has collaborated closely with the State of Wisconsin regarding Math standards in K-12 during 2008-12.
